For (m,nge 1) let (underline{u}= x_1, ldots, x_m ) and (underline{v}=y_1, ldots, y_n) be sequences of primitive elements in a bialgebra and write (u=((x_1x_2)ldots )x_m) and (v= y_1y_2)ldots )y_n) for the corresponding left-normed products.
All the basic notions such as random normed modules, random inner product modules and random locally convex modules, together with their random conjugate spaces, were naturally presented by Guo in the course of the development of random functional analysis [1 4].
Guo initiated a new approach to random functional analysis [1 3], whose main idea is to develop random functional analysis as functional analysis based on random normed modules, random inner product modules and random locally convex modules.
It is known (see [12, 13]) that a semi-inner-product space is a normed linear space with the norm and that every Banach space can be endowed with a semi-inner-product (and in general in infinitely many different ways, but a Hilbert space in a unique way).
It was shown in [1] that a normed space is an inner product space if and only if it is a Ptolemy space.
The Tapia semi-product on the normed space X (see [21]) is the function ((cdot,cdot)_{T}:Xtimes Xtomathbb{R}), defined by (x,y)_{T}=lim_{tsearrow0}frac{varphi(x+ty -varphi(x+ty -varphie (varphi(x)=frac{1}{t}Vert xVert ^{2}), (xin X).
Vector spaces endowed with such data are known as normed vector spaces and inner product spaces, respectively.
Let be a normed space with norm.
It is proven that the Trotter product formula converges in the norm of a symmetrically normed ideal of compact operators away from t0>0 if the Kac operator (the transfer matrix) F t)=e−tB/2e−tAe−tB/2 belongs to this ideal for t=t0.
The notion of orthogonality has many forms when the underlying space is transferred from inner product spaces to real normed spaces.
